Presentation of Combinatorial Concepts in Mathematics Textbooks and Its Compliance with a Concept Development Theory
Abstract
This article analyses the approach taken by five Czech secondary school mathematics textbooks to selected combinatorial concepts, in order to determine the extent to which they provide pedagogical support to teachers based on the theory of generic models, which is a theory of concept development. The analysis of textbooks in relation to this theory focused on the presence and quality of a) isolated models and non-models of future knowledge, b) prompts to generalise as a prerequisite for the creation of a generic model, and c) the supportive role of graphical representations in developing combinatorial thinking.
Most notably, we identified insufficient motivation for combinatorial problems, few isolated models of future knowledge, the absence of explicit prompts to generalise and a consequent lack of a significant concept of isomorphism. Despite the research-proven positive influence of the creation of graphical representations on the development of pupils' combinatorial thinking, they are rare in textbook chapters about combinatorics, and lack diversity.
With a few exceptions, textbook authors do not encourage readers to create their own graphical representations. One textbook stood out in that it frequently prompts the creation of personalised representations, works purposefully with isomorphic problems and encourages the reader to generalise specific procedures.